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The Discontinuous Galerkin Finite Element Method For Burgers Equation

Posted on:2011-04-30Degree:MasterType:Thesis
Country:ChinaCandidate:S ShaoFull Text:PDF
GTID:2120360305987396Subject:Computational Mathematics
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Burgers equation is an important and fundamental partial di?erential equation in ?owmechanics and has found widely used in turbulence, heat conduction, mass transfer, gasdynamics, air and water pollution, continuous stochastic process and so on. This problemcan form shock which makes it di?cult to solve. Burgers equation can be seen as theSimplified model of Navier-Stokes equation, and we can analyze Navier-Stokes equation bysolving the Burgers equation. The research of Burgers equation can make contribution tothe study of many other nonlinear equations. Because Burgers equation can construct theprecise solution, this provides a test model for many nonlinear problems. Therefore, theresearch of numerical method for solving the Burgers equation have important theoreticaland realistic meanings.The first discontinuous Galerkin method was introduced in 1973 by Reed and Hillin the framework of neutron transport. Discontinuous Galerkin methods are a class offinite element methods using completely discontinuous basis functions, which are usuallychosen as piecewise polynomials. Since the basis functions are completely discontinuous,these method have the flexibility which is not shared by typical finite element methods,such as the allowance of arbitrary triangulation with hanging nodes, complete freedomin changing the polynomial degrees in each element independent of that in the neighbors(p-adaptivity). The DG method has found rapid applications in such diverse areas asaeroacoustics, electro-magnetism, gas dynamics, granular ?ows, magneto-hydrodynamics,meteorology, modeling of shallow water, oceanography, weather forecasting, turbulent?ows, turbomachinery and so on.The local discontinuous Galerkin finite element method is an extension of the RKDGmethods to convection-di?usion problems proposed first by Bassi and Rebay in the contextof the compressible Navier-Stokes and recently extended to general convection-diffusionproblems by Cockburn and Shu.This paper consists four chapters. Chapter 1 is preface. We introduce researchbackground, purpose and significance, and describe the research situation of numericalsolution for Burgers equation. Finally, the organizational structure of this paper is given. In chapter 2, we study the local discontinuous Galerkin (LDG) finite element methodfor solving one-dimensional nonlinear Burgers equation with Dirichlet boundary condi-tions. Based on the Hopf-Cole transformation, we transform the original problem intoa linear heat equation with Neumann boundary conditions. The heat equation is thensolved by the LDG finite element method with special chosen numerical ?ux. Theoreticalanalysis show that this method is stable and can achieve arbitrary order of convergencerate. (k + 1)th order of convergence rate is attained when the polynomials Pk are used.Finally, we present some examples of Pk polynomials with 1≤k≤4 to demonstratethe high-order accuracy of this method. The numerical results are also shown to bemore accurate than some available results given in the literature. Then we solve theone-dimensional Burgers equation directly and compare with the indirect solving methodproposed above. Theoretical analysis and numerical experiments show that the indirectsolving method is flexible and need less CPU time.In chapter 3, we solve the two-dimensional Burgers equation by RKDG method. Weuse the generalized slope limiter, rectangular element and uniform mesh grid. We also givethe implement details as well as the example and its numerical results. Comparing withthe triangular element, method using rectangular element need less degree of freedom. Soit is easy to carry out but can attain no bad results.Chapter 4 is conclusion. We compare the indirect solving method with some othernumerical methods for the Burgers equation. Theoretical analysis and numerical exper-iments show that the indirect solving method has higher order of convergence rate. Wealso compare the direct solving method with the indirect solving method. We find that theindirect solving method (can achieve arbitrary order of convergence rate) is more flexiblethan the direct solving method which can attain at most three order of convergence rate.It is shown that the indirect solving method is an e?ective method for Burgers equation.
Keywords/Search Tags:Burgers equation, LDG finite element method, RKDG finite elementmethod, Hopf-Cole transformation, Numerical flux
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